Analytic models of accretion discs - davidar/scholarpedia GitHub Wiki
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3.1. Thin discs 3.2. Thick discs |
Most of the accretion discs types (except proto-planetary and GRB ones) have a negligible self-gravity: the external gravity of the central accreting object dominates. The external gravity is important in shaping several crucial aspects of the internal physics of accretion discs, including their characteristic frequencies (that are connected to several important timescales) and their size (inner and outer radius). The most fundamental gravity's characteristic frequencies are the Keplerian orbital frequency <math>\Omega_K\ ,</math> the radial epicyclic frequency <math>\omega_R\ ,</math> and the vertical epicyclic frequency <math>\omega_Z\ .</math> They are directly relevant for motion of free particles and also play a role for determining equilibria and stability of rotating fluids. In both Newton's and Einstein's gravity the three frequencies are derived from the effective potential <math>U_{eff}(R, j)\ ,</math> and given by the same formulae,
where <math>j</math> is the specific angular momentum, and derivatives are taken at the symmetry plane <math>Z = 0\ .</math> Small (epicyclic) oscillations around the circular orbit <math>R = R_0 = const\ ,</math> <math>Z = 0</math> are governed by <math>\delta{\ddot R} + \omega^2_R\,\delta R = 0\ ,</math> <math>\delta{\ddot Z} + \omega^2_Z\,\delta Z = 0\ ,</math> with solutions <math>\delta{R} \sim \exp( -i\omega_R t)\ ,</math> <math>\delta{Z} \sim \exp( -i\omega_Z t)\ ,</math> which are unstable when <math>\omega^2_R < 0</math> or <math>\omega^2_Z < 0\ .</math>
In Newton's gravity <math>U_{eff} = \Phi + j^2/2R\ .</math> A spherical Newtonian body has the gravitational potential <math>\Phi = -GM/R\ .</math> Thus, in this case, <math>\Omega_K^2 = \omega_R^2 = \omega_Z^2 = GM/R^3 > 0\ ,</math> i.e. all slightly non-circular orbits are closed and all circular orbits are stable. In Einstein's gravity, for a spherical body it is <math>\Omega_K^2 = \omega_Z^2 > \omega_R^2\ ,</math> i.e. non-circular orbits are not closed. In addition, for circular orbits with radii smaller than <math>6GM/c^2\ ,</math> it is <math>\omega_R^2 < 0\ ,</math> which indicates the dynamical instability of these orbits.
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<math>(3.1)</math> |
In the black hole gravity, the stable circular Keplerian orbits exist only with radii greater than the radius of ISCO, or the innermost stable circular orbit radius. All Keplerian orbits closer to the black hole than ISCO are unstable: without an extra support by non-gravitational forces (i.e. pressure or magnetic field) matter cannot stay there orbiting freely, but instead it must fall down into the black hole. This strong-field property of Einstein's gravity, absent in Newton's theory, is the most important physical effect in the black hole accretion disc physics.
The black hole gravitational field is described by three parameters: mass <math>M\ ,</math> angular momentum <math>J</math> and charge <math>Q\ .</math> It is convincingly argued that the astrophysical black holes relevant for accretion discs are uncharged, <math>Q = 0\ .</math> They are described by the stationary and axially symmetric Kerr geometry, with the metric <math>g_{\mu\nu}</math> given in the spherical Boyer-Lindquist coordinates <math>t, \phi, r, \theta</math> by the explicitly known functions of the radius <math>r</math> and the polar angle <math>\theta\ ,</math> which are listed in the table below. The table also gives the contravariant form of the metric, <math>g^{\mu\nu}\ ,</math> defined by <math>g^{\mu\beta}\,g_{\nu\beta} = \delta^{\mu}_{~\nu}\ .</math> It is defined, <math>\Delta = r^2 - 2Mr + a^2\ ,</math> <math>\Sigma = r^2 + a^2\cos^2\theta\ .</math> The signature <math>(+\,-\,-\,-)</math> is used.
The mass and angular momentum have been rescaled into the <math>c = G = 1</math> units, <math>M \rightarrow GM/c^2\ ,</math> <math>J \rightarrow a = J/c\ .</math> For a proper black hole solution it must be <math>\vert a \vert \le M\ ,</math> and the metric with <math>\vert a \vert > M</math> corresponds to a naked singularity. The Penrose cosmic censor hypothesis (unproved) states that there are no naked singularities in the Universe.
| <math>g_{\mu\nu}</math> | |
<math>g^{\mu\nu}</math> | |||||||
| <math>t</math> | <math>\phi</math> | <math>r</math> | <math>\theta</math> | <math>t</math> | <math>\phi</math> | <math>r</math> | <math>\theta</math> | ||
| <math>t</math> | <math>1 - 2\,M\,r/\Sigma</math> | <math>4\,M\,a\,r\sin^2\theta/\Sigma</math> | <math>0</math> | <math>0</math> | |
<math>(r^2 + a^2)^2/\Sigma\,\Delta</math> <math>- a^2\Delta\sin^2\theta/\Sigma\,\Delta</math> |
<math>2M\,\,a\,r/\Sigma\,\Delta</math> | <math>0</math> | <math>0</math> |
| <math>\phi</math> | <math>4\,M\,a\,r\sin^2\theta/\Sigma</math> |
<math>-(r^2 + a^2)\sin^2\theta</math> <math>-2\,M\,a^2r\sin^4\theta/\Sigma</math> |
<math>0</math> | <math>0</math> | |
<math>2M\,\,a\,r/\Sigma\,\Delta</math> | <math>-\frac{\Delta-a^2\sin^2\theta}{\Delta\Sigma\sin^2\theta}</math> | <math>0</math> | <math>0</math> |
| <math>r</math> | <math>0</math> | <math>0</math> | <math>-\Sigma/\Delta</math> | <math>0</math> | |
<math>0</math> | <math>0</math> | <math>-\Delta/\Sigma</math> | <math>0</math> |
| <math>\theta</math> | <math>0</math> | <math>0</math> | <math>0</math> | <math>-\Sigma</math> | |
<math>0</math> | <math>0</math> | <math>0</math> | <math>-1/\Sigma</math> |
In any stationary and axially symmetric spacetime, and in particular in the Kerr geometry, for matter rotating on circular orbits with four velocity <math>u^{\nu} = (u^t, u^{\phi})</math> it is <math>\Omega = u^{\phi}/u^t</math> and <math>j = - u_{\phi}/u_t\ ,</math> from which (and <math>u^{\nu}\,u_{\nu} = 1 </math>) it follows that,
- <math>
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<math>(3.3)</math> |